Let $X$ be the random variable that counts the number of tosses until we get two successive tosses of the same type. Although you did not say so explicitly, I assume that you want the probability mass function of $X$.
What is the probability that $X>n-1$? We need to have a sequence of tosses of length $n-1$ and of type HTHTHT and so on (alternating heads and tails, starting with head), or THTHTH and so on. The probability that the first type of sequence is followed up to and including the $(n-1)$-th toss is $\frac{1}{2^{n-1}}$. The second type of sequence has the same probability, for a total of $\frac{2}{2^{n-1}}$.
The number $X$ of tosses is equal to $n$ if $X>n-1$ and the last toss matches the next to last toss. The probability of that is $\frac{1}{2}$, so
$$P(X=n) =\frac{2}{2^{n-1}}\cdot\frac{1}{2}.$$
This simplifies to $\dfrac{1}{2^{n-1}}$. Note that $n$ ranges over the integers that are $\ge 2$.
Another way: The first toss doesn't matter. After that, it is just a matter of matching the previous toss. So it is very much like tossing a fair coin and winning if you get (say) heads. The only difference is the "wasted" toss at the beginning. If we are tossing a fair coin, and $Y$ is the number of tosses until the first head, then $P(Y=n)=\frac{1}{2^n}$ ($n=1, 2, \dots$). But $X$ has the same distribution as $Y+1$. So $P(X=n+1)=P(Y=n)=\frac{1}{2^n}$, or equivalently $$P(X=n)=P(Y=n-1)=\frac{1}{2^{n-1}}\quad (n=2,3,\dots).$$
Generalization: Suppose that the probability of a head is $p$, and the probability of a tail is $q=1-p$. Either analysis of the problem can be extended to the more general situation. We win on the $n$-th toss if we did not win on the first $n-1$, and then the $n$-th toss matches the previous one.
It is convenient to split the analysis into two cases, $n$ odd and $n$ even. We do a full analysis of the odd case. So let $n=2m$. As is the first analysis, there are two ways to win on the $(2m+1)$-th toss. Either (i) we went HTHTHT and so on, with a tail on the $(2m)$-th toss, and then a tail again on the $(2m+1)$-th toss, or (ii) we went THTHTH and so on, with a head on the $(2m)$-th toss, and a head again on the $(2m+1)$-th toss.
The probability of (i) is $p^mq^m q$ and the probability of (ii) is $q^mp^mp$. Add them together. We get $p^mq^m(p+q)$, which is $p^mq^m$ ($m=1,2,\dots)$.
Next we deal with the case $n$ is even, say $n=2m$. The calculation is very similar to the previous one, so it will be omitted. The probability is the slightly more complicated-looking $p^mq^{m-1}p+q^mp^{m-1}q=p^{m-1}q^{m-1}(p^2+q^2)$, where $m$ ranges over the positive integers.
A simple recursion: Let $E$ be the answer you want and let $E_1$ be the expected number assuming you have tossed at least once (but have not yet won). then $$E=E_1+1$$
as tossing the coin once gets you to the state governed by $E_1$. But then tossing again either ends the game (if you get a match) or leaves you with expectation $E_1$ again. Thus $$E_1=\frac 12 1 \;+\; \frac 12(E_1+1)$$
this is easily solved to get $E_1=2$, whence $E=3$.
Best Answer
Let $p$ be the probability you want.
Look at the possible sequences of three tosses (including the initial $H$). We have $\{HH*,HTT,HTH\}$ Conditional on the first being $H$, the probability of $HH*$ is $\frac 12$, and the probability of the other two are each $\frac 14$. The first is a win for $H$, the second is a win for $T$, and the third restarts the game. Thus $$p=\frac 12\times 0+\frac 14 \times 1 +\frac 14\times p\implies p=\frac 13$$
Note: if you want to allow for a weighted coin, the same calculation works but we need to correct for those initial probabilities. Thus assume that the coin comes up $H$ with probability $\phi$. Then $HH*$ has probability $\phi$, $HTT$ has probability $(1-\phi)^2$, and $HTH$ has probability $(1-\phi)\phi$. So the recursion is now $$p=\phi \times 0+ (1-\phi)^2 \times 1 + (1-\phi)\phi \times p\implies p=\frac {(1-\phi)^2}{1-(1-\phi)\phi}$$ Comparison shows that this matches the result obtained by looking at infinite sums, as in the posted solution of @BCLC (with different notation, unfortunately).