The syntax of `\let` is `\let<token 1><token 2>` where `<token 1>` should be a control sequence or active character. It assigns to `<token 1>` the same meaning `<token 2>` currently has (there are some specialties regarding equals signs and space tokens, something we don't have to care for here).

As a result, to save the current meaning of `\cite` inside another macro you should use `\let<new-macro>\cite` (here `\let\oldcite\cite`). However for macros created with LaTeX2e measurements for robustness (or for optional arguments) you shouldn't use just `\let` but instead should use the newish `\NewCommandCopy` (if that isn't available because of an out of date installation you could use `\usepackage{letltxmacro}` and then `\LetLtxMacro` instead).

With all this your MWE becomes:

```
\documentclass{article}
\NewCommandCopy\oldcite\cite
\usepackage{biblatex}

\begin{document}
See: \oldcite{abcd}

See: \oldcite{efgh}

\begin{thebibliography}{1}
\bibitem{abcd}
\bibitem{efgh}
\end{thebibliography}
\end{document}
```

The reason why you shouldn't just use `\let` is best understood when we look at the actual definition of `\cite` (with the default definition of LaTeX2e):

```
> \cite=macro:
->\protect \cite  .
```

Notice that there are two spaces after the `\cite` in the second line, that's because that macro is actually built with `\csname cite\space\endcsname` (so the name really contains a single trailing space). The real definition sits inside that macro, and `\cite` is just a front facing macro serving to protect `\cite` from further expanding inside many of LaTeX2e's internal processing steps (think `\protected@edef` and the like) and in file output (because TeX expands tokens while writing, e.g., to the ToC, but we typically don't want to expand things just yet but only when the ToC is read back in on the next run).

So when you do `\let\oldcite\cite` what happens is that `\oldcite` gets the meaning `\protect\cite␣` (with trailing space), but it doesn't save the real definition. Now if some package overwrites `\cite` with the same mechanisms used by LaTeX2e (introduced before `\protected` from the e-TeX extension was available) your `\oldcite` doesn't call that definition but something potentially arbitrary.

You're just lucky that `biblatex` is relatively modern and doesn't use that mechanism of protecting but instead uses `\protected`, so get away with it here.

Macros like `\NewCommandCopy` or `\LetLtxMacro` parse the structure of the macro and automagically detect this protection mechanism and do the right thing. So after `\NewCommandCopy\oldcite\cite` you'll get the following definition of `\oldcite`:

```
> \oldcite=macro:
->\protect \oldcite  .
```

and `\oldcite␣` will hold the definition of `\cite␣` for you.