A not quite theta not quite basic hypergeometric function

Had you looked at the modern definition of a $q$-hypergeometric function (from Gasper/Rahman, as noted in the previous answer), you would see that if the numerator and denominator parameters are equal, then there certainly is a $(-1)^n q^\binom{n}{2}$ factor in the sum's general term. Thus, with that definition, your function can be expressed as

$${}_1 \phi_1 \left({{q^{t+1}}\atop{0}}\middle|q;z\right)$$

Let's observe that $[n]_q!=1(1+q)(1+q+q^2)\cdots(1+q+\cdots+q^{n-1})=\frac{(q)_n}{(1-q)^n}$ so that \begin{align} \sum_{n\geq0}(-1)^nq^{\binom{n}2}z^n\frac{[n+t]_q!}{[n]_q![t]_q!} &=\sum_{n\geq0}(-1)^nq^{\binom{n}2}z^n\frac{(q)_{n+t}}{(q)_n(q)_t} \\ &=\sum_{n\geq0}(-1)^nq^{\binom{n}2}z^n\frac{(q^{t+1})_n}{(q)_n} \\ &=\lim_{\tau\rightarrow0}\,\,\, {}_2 \phi_1 \left({{\frac1{\tau},q^{t+1}}\atop{\tau}}\middle|\,\,q,z\tau\right) \\ &=\lim_{\tau\rightarrow0}\,\,\frac{(q^{t+1})_{\infty}(z)_{\infty}}{(\tau)_{\infty}(z\tau)_{\infty}}\,\,\cdot\,\, {}_2 \phi_1 \left({{\frac{\tau}{q^{t+1}},z\tau}\atop{z}}\middle|\,\,q,q^{t+1}\right) \\ &=(q^{t+1})_{\infty}(z)_{\infty}\,\sum_{n\geq0}\frac{q^{n(t+1)}}{(q)_n(z)_n}; \end{align} where we have applied a Heine transformation in the penultimate step.

Perhaps other Heine transformations would lead to something more interesting.

"Basic hypergeometric series are series $\sum c_n$, with $c_{n+1}/c_n$ a rational function of $q^n$, for a fixed parameter $q$, which is usually taken to satisfy $|q|<1$, but at other times is a power of a prime."

This quote is from the Forward (by Richard Askey) to "Basic Hypergeometric Series", by Gaspar and Rahman, p. xvi, which can be found on the preview here: https://www.amazon.com/Basic-Hypergeometric-Encyclopedia-Mathematics-Applications/dp/0521833574/ref=sr_1_1?ie=UTF8&qid=1495081420&sr=8-1&keywords=basic+hypergeometric+series

Your series above, as well as theta series, are both basic hypergeometric series.

There are different conventions for their notation; some such as Gasper and Rahman's notation make (powers of) the factor $(-1)^n q^{(n-1)n/2}$ explicit for some cases, but not others. In any event, regardless of the notation, these factors arise by changing variables and considering special cases. For example, the series in your question arises from Heine's series $$ _2\phi_1(a,b,c;q;z) = \sum_{n\geq 0} \frac{(a)_n(b)_n}{(c)_n(q)_n}z^n $$ by setting $a=q^{1+t}$, $c=0$, making the change of variables $b\mapsto 1/b$, then $z\mapsto bz$, and finally setting $b=0$.

The book linked above is essentially the bible for these series.